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A general structure theorem for the nash equilibrium correspondence

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  • Predtetchinski, A.

    (Microeconomics & Public Economics)

Abstract

I consider n-person normal form games where the strategy set of each player is a non-empty compact convex subset of an Euclidean space, and the payoff function of player i is continuous in joint strategies and continuously differentiable and concave in the player i's strategy. No further restrictions (such as multilinearity of the payoff functions or the requirement that the strategy sets be polyhedral) are imposed. I demonstrate that the graph of the Nash equilibrium correspondence on this domain is homeomorphic to the space of games. This result generalizes a well-known structure theorem in [Kohlberg, E., Mertens, J.-F., 1986. On the strategic stability of equilibria. Econometrica 54, 1003-1037]. It is supplemented by an extension analogous to the unknottedness theorems in [Demichelis S., Germano, F., 2000. Some consequences of the unknottedness of the Walras correspondence. J. Math. Econ. 34, 537-545; Demichelis S., Germano, F., 2002. On (un)knots and dynamics in games. Games Econ. Behav. 41, 46-60]: the graph of the Nash equilibrium correspondence is ambient isotopic to a trivial copy of the space of games.
(This abstract was borrowed from another version of this item.)
(This abstract was borrowed from another version of this item.)

Suggested Citation

  • Predtetchinski, A., 2006. "A general structure theorem for the nash equilibrium correspondence," Research Memorandum 010, Maastricht University, Maastricht Research School of Economics of Technology and Organization (METEOR).
  • Handle: RePEc:unm:umamet:2006010
    DOI: 10.26481/umamet.2006010
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    References listed on IDEAS

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    1. Blume, Lawrence E & Zame, William R, 1994. "The Algebraic Geometry of Perfect and Sequential Equilibrium," Econometrica, Econometric Society, vol. 62(4), pages 783-794, July.
    2. Govindan, Srihari & Wilson, Robert, 2001. "Direct Proofs of Generic Finiteness of Nash Equilibrium Outcomes," Econometrica, Econometric Society, vol. 69(3), pages 765-769, May.
    3. Kohlberg, Elon & Mertens, Jean-Francois, 1986. "On the Strategic Stability of Equilibria," Econometrica, Econometric Society, vol. 54(5), pages 1003-1037, September.
    4. DeMichelis, Stefano & Germano, Fabrizio, 2000. "Some consequences of the unknottedness of the Walras correspondence," Journal of Mathematical Economics, Elsevier, vol. 34(4), pages 537-545, December.
    5. Stefano Demichelis & Klaus Ritzberger & Jeroen M. Swinkels, 2004. "The simple geometry of perfect information games," International Journal of Game Theory, Springer;Game Theory Society, vol. 32(3), pages 315-338, June.
    6. Balasko, Yves, 1978. "Economic Equilibrium and Catastrophe Theory: An Introduction," Econometrica, Econometric Society, vol. 46(3), pages 557-569, May.
    7. Demichelis, Stefano & Germano, Fabrizio, 2002. "On (un)knots and dynamics in games," Games and Economic Behavior, Elsevier, vol. 41(1), pages 46-60, October.
    8. Zhou, Yuqing, 1997. "Genericity Analysis on the Pseudo-Equilibrium Manifold," Journal of Economic Theory, Elsevier, vol. 73(1), pages 79-92, March.
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    Cited by:

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    3. Pahl, Lucas, 2023. "Polytope-form games and index/degree theories for extensive-form games," Games and Economic Behavior, Elsevier, vol. 141(C), pages 444-471.
    4. Philippe Bich & Julien Fixary, 2021. "Oddness of the number of Nash equilibria: the Case of Polynomial Payoff Functions," Documents de travail du Centre d'Economie de la Sorbonne 21027, Université Panthéon-Sorbonne (Paris 1), Centre d'Economie de la Sorbonne.
    5. Philippe Bich & Julien Fixary, 2021. "Structure and oddness theorems for pairwise stable networks," Post-Print halshs-03287524, HAL.
    6. Philippe Bich & Julien Fixary, 2021. "Oddness of the number of Nash equilibria: the case of polynomial payoff functions," Post-Print halshs-03354269, HAL.
    7. Philippe Bich & Julien Fixary, 2021. "Oddness of the number of Nash equilibria: the case of polynomial payoff functions," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-03354269, HAL.

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